Finite Group Databasebeta

← #16856166

Subgroups — #16856166

Order 3968.

552 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 153 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758486C211yesmaximal, characteristic, complemented
2C2×#11758519C211yesmaximal, characteristic, complemented
3C2×#11758528C211yesmaximal, characteristic, complemented
4#14521595C211yesmaximal, characteristic
5C2×#11758521C211yesmaximal, characteristic, complemented
6#14521442C212yesmaximal, complemented
7#14521440C212yesmaximal, complemented
8#14521427C212yesmaximal, complemented
9#14521420C212yesmaximal, complemented
10M16×Dic31C211yesmaximal, characteristic, complemented
11C2×C31×#137C211yesmaximal, characteristic, complemented
12C2×#41×C31C2211yescharacteristic
13C22×Q8×C31C2211yescharacteristic
14C2×#92495C2211yescharacteristic
15C22×Dic62C2211yescharacteristic
16C31×#137—24no—
17C2×Q16×C31C2212yes—
18C2×SD16×C31C2212yes—
19C2×M16×C31C2211yescharacteristic
20#11758528—24no—
21#11758524C2211yescharacteristic
22#11758521—22no—
23#11758521C2212yes—
24#11758521C2211yescharacteristic
25#11758521C2212yes—
26C2×#92482C2212yes—
27C2×#92482C2212yes—
28#11758519—24no—
29C2×#92481C2212yes—
30#11758517C2211yescharacteristic
31#11758516—22no—
32#11758513C2211yescharacteristic
33C2×#92480C2212yes—
34C4×C22⋊D31—22no—
35C2×Dic31⋊C4—22no—
36C2×C4×Dic31C2211yescharacteristic
37C2×#92474C2211yescharacteristic
38#11758486—24no—
39C2×Dic124C2212yes—
40C2×#92470C2212yes—
41#11758403C2212yes—
42C8×Dic31C2212yes—
43#41×C31—22no—
44#41×C31—22no—
45C2×Q8×C31—21no—
46C2×Q8×C31—22no—
47C2×Q8×C31—22no—
48C2×Q8×C31C2312yes—
49C2×Q8×C31C2311yescharacteristic
50C2×D4×C31—21no—
51C2×D4×C31C2311yescharacteristic
52C22×C4×C31—21no—
53C22×C4×C31—21no—
54C22×C4×C31C2311yescharacteristic
55C2×C22⋊D31—21no—
56C2×C22⋊D31—22no—
57C22×Dic31—21no—
58C22×Dic31D412yes—
59#92495—22no—
60#92495—22no—
61C2×D124C2311yescharacteristic
62C2×C4×D31—21no—
63C2×Dic62—22no—
64C2×Dic62—22no—
65C2×Dic62—21no—
66C2×Dic62C2312yes—
67C2×Dic62C2311yescharacteristic
68Q16×C31—24no—
69SD16×C31—24no—
70M16×C31D414yescomplemented
71C2×C8×C31C2312yes—
72C22⋊Dic31—22no—
73#92482—24no—
74#92482—24no—
75#92481—24no—
76#92480—24no—
77(C2×D31)⋊C4—22no—
78Dic31⋊C4—24no—
79Dic31⋊C4—24no—
80Dic31⋊C4—22no—
81C4×Dic31D412yes—
82C4×Dic31C2311yescharacteristic
83C4×Dic31C2311yescharacteristic
84#92474D414yescomplemented
85C2×C31⋊C8C2312yes—
86Dic124—24no—
87#92470—24no—
88C23×C31—21no—
89C23×C31C2×D411yescharacteristic
90C22×D31—21no—
91Q8×C31—22no—
92Q8×C31—21no—
93Q8×C31—22no—
94Q8×C31—21no—
95Q8×C31—42no—
96D4×C31—21no—
97D4×C31—21no—
98D4×C31—42no—
99C2×C4×C31—22no—
100C2×C4×C31—21no—
101C2×C4×C31—42no—
102C2×C4×C31—42no—
103C2×C4×C31C2×D411yescharacteristic
104C2×C4×C31C2×D412yes—
105C2×C4×C31C2×D412yes—
106C2×C4×C31C2411yescharacteristic
107C22⋊D31—42no—
108C22⋊D31—44no—
109C2×Dic31—24no—
110C2×Dic31—22no—
111C2×Dic31—21no—
112C2×Dic31—42no—
113C2×Dic31C2×D412yes—
114C2×Dic31C2×D412yes—
115D124—21no—
116D124—21no—
117C4×D31—42no—
118Dic62—22no—
119Dic62—21no—
120Dic62—22no—
121Dic62—21no—
122Dic62—42no—
123C8×C31C2×D414yes—
124C31⋊C8C2×D414yes—
125#3152—311nomaximal, Sylow
126C22×C31—22no—
127C22×C31—41no—
128C22×C31—42no—
129C22×C31#12712yes—
130C22×C31C22×D411yescharacteristic
131C2×D31—41no—
132C2×D31—42no—
133C4×C31—41no—
134C4×C31—42no—
135C4×C31#12712yes—
136C4×C31C22×D411yescharacteristic
137C4×C31C22×D411yescharacteristic
138Dic31—22no—
139Dic31—42no—
140Dic31—41no—
141Dic31#13714yescomplemented
142C2×#137—311no—
143C2×#137—311no—
144C2×#137—311no—
145C2×#137D3111yescharacteristic, complemented
146#532—311no—
147C2×C4⋊Q8—311no—
148#497—312no—
149#497—312no—
150#494—312no—
151#492—312no—
152C4×M16—311no—
153C2×C31—22no—
154C2×C31—81no—
155C2×C31C2×#12711yescharacteristic
156C2×C31C2×#13712yes—
157D31—81no—
158C2×#41—311no—
159C2×#41C2×D3111yescharacteristic
160C22×Q8—311no—
161C22×Q8C2×D3111yescharacteristic
162#137—24no—
163#137—624no—
164#137—624no—
165#137—624no—
166C2×Q16—312no—
167C2×Q16—312no—
168C2×Q16—312no—
169C2×Q16C2×D3112yes—
170C2×SD16—312no—
171C2×SD16—312no—
172C2×SD16—312no—
173C2×SD16C2×D3112yes—
174C2×M16—311no—
175C2×M16C2×D3111yescharacteristic
176C4⋊Q8—312no—
177C4⋊Q8—312no—
178C4⋊Q8—311no—
179C4⋊Q8—622no—
180#127—311no—
181#124—311no—
182#124—311no—
183#121—622no—
184C4×D4—622no—
185C2×C4⋊C4—622no—
186C2×C42—311no—
187#97—312no—
188C4×C8—312no—
189C31#315211yescharacteristic, complemented, Sylow
190#41—22no—
191#41—22no—
192#41—622no—
193#41—622no—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—311no—
198C2×Q8—312no—
199C2×Q8—621no—
200C2×Q8—622no—
201C2×Q8—622no—
202C2×Q8C22×D3111yescharacteristic
203C2×Q8C22×D3112yes—
204C2×D4—21no—
205C2×D4—311no—
206C2×D4—622no—
207C2×D4—621no—
208C2×D4C22×D3111yescharacteristic
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—312no—
212C22×C4—621no—
213C22×C4—621no—
214C22×C4C22×D3111yescharacteristic
215Q16—24no—
216Q16—624no—
217Q16—624no—
218Q16—624no—
219SD16—24no—
220SD16—624no—
221SD16—624no—
222SD16—624no—
223M16—314no—
224M16C22⋊D3114yescomplemented
225C2×C8—312no—
226C2×C8C22×D3112yes—
227C4⋊C4—624no—
228C4⋊C4—624no—
229C4⋊C4—622no—
230C22⋊C4—622no—
231C22⋊C4—622no—
232C42—312no—
233C42—311no—
234C42—311no—
235C23—21no—
236C23—621no—
237C23D4×D3111yescharacteristic
238Q8—22no—
239Q8—21no—
240Q8—22no—
241Q8—21no—
242Q8—42no—
243Q8—622no—
244Q8—621no—
245Q8—622no—
246Q8—621no—
247Q8—1242no—
248D4—21no—
249D4—21no—
250D4—42no—
251D4—621no—
252D4—621no—
253D4—1244no—
254D4—1242no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—312no—
260C2×C4—312no—
261C2×C4—624no—
262C2×C4—621no—
263C2×C4—622no—
264C2×C4—1242no—
265C2×C4—1242no—
266C2×C4D4×D3111yescharacteristic
267C2×C4D4×D3112yes—
268C2×C4C2×C22⋊D3112yes—
269C2×C4C23×D3111yescharacteristic
270C8—314no—
271C8C2×C22⋊D3114yes—
272C22—22no—
273C22—42no—
274C22—41no—
275C22—1242no—
276C22—1241no—
277C22#1175851712yes—
278C22C2×D4×D3111yescharacteristic
279C4—41no—
280C4—42no—
281C4—314no—
282C4—622no—
283C4—1241no—
284C4—1242no—
285C4#1175851712yes—
286C4C2×D4×D3111yescharacteristic
287C4C22×C22⋊D3111yescharacteristic
288C2—22no—
289C2—81no—
290C2—2481no—
291C2C2×#1175851711yescharacteristic
292C2#137×D3112yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×#137C2
socleC22×C31C22×D4
prime radical#14521595C2
hypercenterC2×#137D31
hypocenterC31#3152
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#11758486C2semidirect
2. C2×#11758519C2semidirect
3. C2×#11758528C2semidirect
5. C2×#11758521C2semidirect
6. #14521442C2semidirect
7. #14521440C2semidirect
8. #14521427C2semidirect
9. #14521420C2semidirect
10. M16×Dic31C2semidirect
11. C2×C31×#137C2semidirect
70. M16×C31D4semidirect
84. #92474D4semidirect
141. Dic31#137semidirect
145. C2×#137D31semidirect
189. C31#3152semidirect
224. M16C22⋊D31semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#117584862
C2×#117585192
C2×#117585282
#145215952
C2×#117585212
#145214422
#145214402
#145214272
#145214202
M16×Dic312
C2×C31×#1372
#315231